PRECALCULUS: Transformations of Functions Speed Dating

I know I am going to take some grief for this statement, but I have learned it to be the truth. There are some things that, once discovered, you simply need to memorize — and practice. This is about mental (and often physical) motor skills. Becoming fluent at some things requires more or less practice for different people, but for most of us, it does require something.

I come by this knowledge honestly.

I have played the piano since I was three or four. In growing as a pianist, there are conceptual learnings, procedural learnings, and contextual learnings. Technique, concepts, and contexts. To play well, you need first to become proficient and fluent technically. That is why there are so many established sets of technical figure exercises: Hanon, Czerny, and many others. The same is true for the violin (Schradieck, Kreutzer, Hrimaly) and every other instrument under the sun. If you want to be capable of playing the advanced works in your instrument's repertoire, your fingers and body need to know how to flow over the keys or strings in every known and familiar situation (scales, arpeggios, finger-crossings) so that you can summon them automatically in your pursuit of unknown and unfamiliar territory in the literature of your instrument.

But working your way through all of the possible finger movements will only make you a master technician. Doing only finger exercises will not give you a very deep understanding of music. It simply builds the finger and muscle memory you will need to face different and difficult technical situations and requirements as you dive deeper and deeper into the piano repertoire.

In other words, technical proficiency is necessary, but not sufficient to become a strong and capable musician. Without technique, you cannot progress.

Beyond technique, there are concepts to master — harmony, counterpoint, and composition, as well as music history. If you don't understand the basics of scales and harmonies and chord progressions in Western music, you will lack the basic musicianship that is required to make sense of the piano repertoire. You need to understand how melodic lines or voices can be woven together using harmonies and rhythms and chord progressions to build a piece of music with coherent and reproducible grammar and syntax that can be both encoded and decoded by others. Without the foundational concepts, your attempts to interpret and play the repertoire will be incoherent.

Finally, there are contexts — historical, cultural, interpretive. The ultimate context is your instrument's repertoire. Music, like mathematics, is a cultural act. As a musical learner, you need to be mentored into the repertoire. You need to experience other musicians' interpretations and experiences to learn what competent, coherent, and ultimately subtle musical communication sounds, feels, and looks like. And while you are doing this, you also need to be able to explore the repertoire for yourself so that you can find your own way.

It seems to me that this is a similar situation to what we face in mathematics education. Technical proficiency is boring but somewhat mindless. If you had to listen to anyone (including yourself) only playing Hanon's same 60 exercises day in and day out, you would undoubtedly lose your mind. As music, the technical patterns are boring — up and down, back and forth, crossing and uncrossing, stretching and shifting. But they're necessary to develop a foundation of muscle memory and motor skills, as well as the habits of mind and of practice you will need as you gain proficiency and advance to building the finer and finer skills of musicianship.

So there was a need for teachers to create pieces that are very rich musically but very restricted technically so that they are accessible to new learners. This is why Bach, for example, wrote the pieces in his Anna Magdalena Notebook. They created an on-ramp for his new wife to be included in the family's deep musical conversations. She needed a low barrier-to-entry, high-ceiling on-ramp to sophisticated music. These are also the motivation behind Bach's Two- and Three-Part Inventions. Bach's life as a composer was inextricably bound up in his life as a teacher, and these accessible works are the "rich problems" of piano education. They are accessible pieces that even the greatest virtuosi find beautiful.

This is why I love Glenn Gould's recordings of Bach's Two- and Three-Part Inventions. They are the music of his childhood, but from an advanced standpoint. The recordings are filled with joy and delight. To the dismay of critics, he often hums along with himself as he plays. The critics find this annoying. Personally, I find it inspiring. If Glenn Gould can still delight in playing these pieces, then so can I. It gives me great permission. The recordings and the playing are magical.

In my view, this is how we should be looking at the balance we need to strike in math education. There are technique and concepts and repertoire that learners need. Without strong technique, your understanding of concepts will be shallow. Without context/repertoire, your understanding and practice of mathematics will be joyless and without wonder.

The long-range purpose
of our practice was clearly
stated on the board

So this is the basis from which I used Kate Nowak's Speed Dating structure to solidify my Precalculus students' early understanding and integration of transformations of functions on Friday. As I keep reminding them, the parent functions and their graphs, as well as the transformations of these basic function graphs, are the essential vocabulary development work for calculus. This is the Hanon and Czerny mindset shift on which we are focusing this year: elementary things that we consider from an advanced standpoint. The order of operations becomes more sophisticated. "Groupings" replace "parentheses" in your thinking about where to start. Groupings, I warn them, are masters of disguise. Sometimes they show up as parentheses, but much of the time they show up wearing a moustache or another costume. They might show up in the form of the absolute value bars or the fraction bar. Don't be fooled, I warn them. Keep your focus on whether you are dealing with transformations of inputs or outputs.

We started simply, using only a single function du jour. On Friday, that was the square root of x. We also restricted our investigation to horizontal and vertical shifts as well as reflections across the x- and y-axis. We were considering the impact on the graphs of basic parent functions as we operated on either the input or the output of the function. We did not multiply by any value other than –1 to start. The Day 1 problem cards are here.  The Day 2 problem cards are also available now (formatted by the amazing Meg Craig - thank you!).

On Friday, each of my 36 students started out with his or her own problem card that contained two related transformations — a shift and a reflection. We organized the speed dating structure, moved our backpacks along the two empty walls, and established our rules of movement (the students along the window side of each row travel, while the hallway-side students stay where they are). If you run into problems, ask the expert in the room on that problem. He or she is sitting directly across from you.

36 precalc students in a state of flow

Trade, analyze, investigate, sketch, discuss. For forty minutes, my Precalculus students lost themselves in analyzing, investigating, sketching, and discussing functional operations on inputs and outputs. Every two minutes, my iPhone timer would go off and I would call out, "Shift!" And all 36 students would trade back their problem cards, while half of them stood up and moved one seat to their right. Then I would reset the timer and they would lose their ego-selves in each new immersion. I loved the hush that fell over the room after everybody settled down into the next round of problems. I eavesdropped on moving and purposeful conversation about inputs and outputs of functions, shifting and reflecting, as they worked collaboratively to help each other and to help themselves attain proficiency. The preciousness of each minute of mathematical conversation was not lost on me.

And I tell you, it was glorious.

WEEK 1 - PROJECT 'MAD MEN' -- Classroom Rules PSA Skits

Leonard Bernstein once said, "To achieve great things, two things are needed — a plan and not quite enough time." I decided to put that principle to the test this first week at my new school by assigning a project on Day 1 that thew together strangers with an absurd but achievable goal: given a particular classroom rule or guideline, create a Public Service Announcement  (i.e., a 30-second "TV commercial" in the form of a skit) whose purpose was to motivate viewers to follow the rules/guidelines for the good of the group.

I created a set-up, instructions, and a rubric for the group project. And my students did not disappoint.

The idea was to get students to think about the consequences of their actions and choices, but their ideas for implementation exceeded even my wildest dreams. Most skits followed a "slice of life" strategy, but the ones that really blew us all away were the ones that parodied existing campaigns.

Two brilliant PSAs started from already-iconic Geico insurance commercials, but the one that left me with tears running down my cheeks was a take-off on Sarah McLachlan's ASPCA spots. The song, "In the arms of the angels..." began playing, and student "Sarah" appears onstage making the exact same kind of appeal she makes in those ads. They had the tone, cadence, and music exactly right, and they clearly understood the emotionally manipulative rhetorical strategy — the seemingly endless list of forms of ignorance designed to eventually provoke self-recognition in almost everyone. Their "mathematical justification" was as follows: the narrator enters and says, "In the past year alone, texting in class tragically cost 5 of Doctor X's students their lives. Remember, think twice before texting in class — there may be fatal consequences for your grade, and for you!

It was pure and inspired genius.

I also loved the spot-on impressions of my teacher persona. One student gave a pitch-perfect parody of my "Function Basics" talk that made me both cringe and laugh my ass off simultaneously.

The best thing about this assignment was that it really pushed the voice of authority downward, into the student community itself. Whatever they made of the experience, they owned it.

I am going to try and remember this for later in the semester, when we've become too routinized.

This is definitely going to be an ongoing part of my repertoire of Day 1 activities. I got through what I neded to,  then gave them the rest of the abbreviated period to collaborate. The time pressure was a thing of art.

It was perfectly imperfect — exactly the way first days ought to be.

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UPDATE:

Here is the link to a generified Word document that you can customize for your own class:

PROJECT MAD MEN- classroom rules PSA generic.doc

And here are the three sample 30-second PSAs I showed my classes to give them ideas:

'You Lost Your Life!' – game show hosted by the Crash Test Dummies (Since Vince & Larry, the Crash Test Dummies, were introduced to the American public in 1985, safety belt usage has increased from 14% to 79%, saving an estimated 85,000 lives, and $3.2 billion in costs to society)
'
What could you buy with the money you save?' - throwing things over a cliff (You could purchase TVs, bicycles, and computers with the money most families spend on wasted electricity)
'Five Seconds' – at highway speeds, the average text takes your eyes off the road for 5 seconds (Five seconds is the average time your eyes are off the road while texting. When traveling at 55mph, that's enough time to cover the length of a football field)

Edwin Moise on postulates considered as self-evident truths

My favorite book I've read this summer has hands-down been Edwin E. Moise's, Elementary Geometry From An Advanced Standpoint (yes, I know, I'm a nerd). I'm nowhere near finished with it, but in planning for teaching Geometry this year, I've used it a lot to help me think through what we assert and why we assert it.

This section is a beautiful summary of the many wonders and problems with an axiomatic system such as the one we continue to teach. I don't know exactly how I'm going to use this, but it is definitely going to inform my presentation of something.

pp. 282-3

27.2 POSTULATES CONSIDERED AS SELF-EVIDENT TRUTHS

In the time of Euclid, and for over two thousand years thereafter, the postulates of geometry were thought of as self-evident truths about physical space; and geometry was thought of as a kind of purely deductive physics. Starting with the truths that were self-evident, geometers considered that they were deducing other and more obscure truths without the possibility of error. (Here, of course, we are not counting the casual errors of individuals, which in mathematics are nearly always corrected rather promptly.) This conception of the enterprise in which geometers were engaged appeared to rest on firmer and firmer ground as the centuries wore on. As the other sciences developed, it became plain that in their earlier stages they had fallen into fundamental errors. Meanwhile the “self-evident truths” of geometry continued to look like truths, and also continued to seem self-evident.

With the development of hyperbolic geometry, however, this view became untenable. We then had two different, and mutually incompatible, systems of geometry. Each of them was mathematically self-consistent, and each of them was compatible with our observations of the physical world. From this point on, the whole discussion of the relation between geometry and physical space was carried on in quiet different terms. We now thing not of a unique, physically “true” geometry, but of a number of mathematical geometries, each of which may be a good or bad approximation of physical space, and each of which may be useful in various physical investigations. Thus we have lost our faith not only in the idea that simple and fundamental truths can be relied upon to be self-evident, but also in the idea that geometry is an aspect of physics.

This philosophical revolution is reflected, oddly enough, in the differences between the early passages of the Declaration of Independence and the Gettysburg Address. Thomas Jefferson wrote:

  “ . . . We hold these truths to be self-evident, that all men are created equal, that they are endowed by their creator with certain unalienable rights, that among these are Life, Liberty and the pursuit of Happiness . . . . “

The spirit of these remarks is Euclidean. From his postulates, Jefferson went on to deduce a nontrivial theorem, to the effect that the American colonies had the right to establish their independence by force of arms.

Lincoln spoke in a quite different style:

  “Fourscore and seven years ago our fathers brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal.”

Here Lincoln is referring to one of the propositions mentioned by Jefferson, but he is not claiming, as Jefferson did, that this proposition is self-evidently true, or even that it is true at all. He refers to it merely as a proposition to which a certain nation was dedicated. Thus, to Lincoln, this proposition is a description of a certain aspect of the United States (and, of course, an aspect of himself). (I am indebted for this observation to Lipman Bers.)

This is not to say that Lincoln was a reader of Lobachevsky, Bolyai or Gauss, or that he was influenced, even at several removes, by people who were. It seems more likely that a shift in philosophy had been developing independently  of the mathematicians, and that this helped to give mathematicians the courage to undertake non-Euclidean investigations and publish the results.

At any rate, modern mathematicians use postulates in the spirit of Lincoln. The question where the postulates are “true” does not even arise. Sets of postulates are regarded merely as descriptions of mathematical structures. Their value consists in the fact that they are practical aids in the study of the mathematical structures that they describe.